where \(\hat{a}^{\rm {in}}_{a^{\prime }}\)  (\(\hat{a}^{\rm {in}}_{b^{\prime }}\) ) denote the annihilation operator of the incident modes \(a^{\prime }\)  (\(b^{\prime }\) ) and obey the commutation relation \([\hat{O}, \hat{O}^{\dagger }]=1\)  (\(\hat{O} = \hat{a}^{\rm {in}}_{a^{\prime }}, \hat{a}^{\rm {in}}_{b^{\prime }}\) ). The transmission and reflection coefficients \(t_{a^{\prime } b}\)  and \(r_{b^{\prime }b}\) , subject to a constraint [1]} \(\sum _{a^{\prime }} |t_{a^{\prime }b}|^2 + \sum _{b^{\prime }} |r_{b^{\prime }b}|^2 = 1\) , can be approximately regarded as complex Gaussian random variables [2]}. Accordingly, \(t_{a^{\prime }b} = \sqrt{T_{a^{\prime }b}} e^{i\phi _{a^{\prime }b}}\)  and \(r_{b^{\prime }b} = \sqrt{R_{b^{\prime }b}} e^{i\phi _{b^{\prime }b}}\)  where \(\phi _{a^{\prime }b}\)  (\(\phi _{b^{\prime }b}\) ) is indeed uniformly distributed in the interval [\(0,2\pi \) ] while \(T_{a^{\prime }b}\)  and \(R_{b^{\prime }b}\)  are the variables of Rayleigh distribution. In addition, the ensemble-averaged transmission and reflection coefficients are given by \(\overline{T_{a^{\prime }b}} = 1/(Ms)\)  and \(\overline{R_{b^{\prime }b}} = (1-1/s)/M\)  [3]}, [4]}, where \(M\)  represents the number of transmission channels and the overline indicates the average over ensembles. It is easily seen that as the disorder strength \(s\)  increases, the average transmission coefficient \(\overline{{T_{a^{\prime }b}}}\)  decreases. It is worth noting that Eq. (REF ) quantifies the very general input-output relation. The specific characteristics of the multiple scattering disordered medium are represented by the reflection and transmission coefficients. For instance, \(t_{a^{\prime }b}\)  describes the coupling between the output mode \(b\)  and the input mode \(a^{\prime }\) .
